Abstract
A regular language [Formula: see text] is non-returning if in the minimal deterministic finite automaton accepting it there are no transitions into the initial state. Eom, Han and Jirásková derived upper bounds on the state complexity of boolean operations and Kleene star, and proved that these bounds are tight using two different binary witnesses. They derived tight upper bounds for concatenation and reversal using three different ternary witnesses. These five witnesses use a total of six different transformations. We show that for each [Formula: see text], there exists a ternary witness of state complexity [Formula: see text] that meets the bound for reversal, and restrictions of this witness to binary alphabets meet the bounds for star, product, and boolean operations. Hence all of these operations can be handled simultaneously with a single witness, using only three different transformations. We also derive tight upper bounds on the state complexity of binary operations that take arguments with different alphabets. We prove that the maximal syntactic semigroup of a non-returning language has [Formula: see text] elements and requires at least [Formula: see text] generators. We find the maximal state complexities of atoms of non-returning languages. We show that there exists a most complex sequence of non-returning languages that meet the bounds for all of these complexity measures. Furthermore, we prove there is a most complex sequence that meets all the bounds using alphabets of minimal size.
Highlights
Formal definitions are postponed until Section 2; we assume the reader is familiar with basic properties of regular languages and finite automata as described in [11, 13], for example
The syntactic semigroup is isomorphic to the transition semigroup of the minimal deterministic finite automaton (DFA) of L, that is, the semigroup of transformations of the state set of the DFA induced by non-empty words
We demonstrate that the upper bound on the state complexity of reversal is met by a single ternary language, and that no binary language meets this bound
Summary
Formal definitions are postponed until Section 2; we assume the reader is familiar with basic properties of regular languages and finite automata as described in [11, 13], for example. The state complexities of common operations (union, intersection, difference, symmetric difference, Kleene star, reverse and product/concatenation) were studied by Eom, Han and Jirásková [7] They pointed out that several interesting subclasses of regular languages have the non-returning property; these subclasses include the class of suffix-free languages (suffix codes) and its subclasses (for example, bifix-free languages), and finite languages. The syntactic semigroup is isomorphic to the transition semigroup of the minimal DFA of L, that is, the semigroup of transformations of the state set of the DFA induced by non-empty words Another complexity measure suggested in [2] is the number and state complexities of the atoms of the language, where an atom is a certain kind of intersection of complemented and uncomplemented quotients of L. Omitted proofs can be found at http://arxiv.org/abs/1701.03944
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